Unit 5, Lesson 9
Using Base-Ten Diagrams to Divide
Grade 6
9.1
Warm-up
Elena used base-ten diagrams to find .
She started by representing 372.
She made 3 groups, each with 1 hundred. Then, she put the tens and ones in each of the 3 groups. Here is her diagram for .
Discuss with a partner:
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Elena’s diagram for 372 has 7 tens. The one for has only 6 tens. Why?
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Where did the extra ones (small squares) come from?
9.2
Activity
Mai used base-ten diagrams to calculate . She started by representing 62.
She then made 5 groups, each with 1 ten. There was 1 ten left. She decomposed it into 10 ones and distributed the ones across the 5 groups.
Here is Mai’s diagram for .
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Discuss these questions with a partner:
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Mai should have a total of 12 ones, but her diagram shows only 10. Why?
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She did not originally have tenths, but in her diagram each group has 4 tenths. Why?
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What value has Mai found for ?
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- Find the quotient of . Show your reasoning. If you get stuck, try drawing a base-ten diagram or using base-ten representations.
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Four students share a \$271 prize from a science competition. How much does each student get if the prize is shared equally? Show your reasoning.
9.3
Activity
To find using diagrams, Elena began by representing 53.8.
She placed 1 ten into each group, decomposed the remaining 1 ten into 10 ones, and went on distributing the units.
This diagram shows Elena’s initial placement of the units and the decomposition of 1 ten.
4 groups of base-ten diagrams. Group 1, 1 rectangle labeled, tens, 1 square labeled, ones, 2 small rectangles labeled, tenths. Group 2, 1 rectangle labeled, tens, 1 square labeled, ones, 2 small rectangles labeled, tenths. Group 3, 1 rectangle labeled, tens, 1 square labeled, ones, 2 small rectangles labeled, tenths. Group 4, 1 rectangle labeled, tens, 2 small rectangles labeled, tenths. Below the groups, one rectangle decomposed into 10 squares.
Here’s Elena’s finished diagram, showing the quotient of .
Discuss with a partner:
- What did Elena do after decomposing the 1 ten into 10 ones? How did she get to the last diagram?
- Based on Elena’s work, what is the value of ?
Student Lesson Summary
One way to find the quotient of two numbers, such as , is to use a base-ten diagram to represent the hundreds, tens, and ones and to create equal-size groups.
We can think of the division by 3 as splitting up 345 into 3 equal groups.
Each group has 1 hundred, 1 ten, and 5 ones, so . Notice that in order to split 345 into 3 equal groups, one of the tens had to be decomposed into 10 ones.
Base-ten diagrams can also help us think about division when the result is not a whole number. Let’s look at , which we can think of as dividing 86 into 4 equal groups.
4 groups of base-ten diagrams. Each group contains 2 rectangles labeled, tens, and 1 square labeled, ones. 2 squares are decomposed into 20 small rectangles labeled, tenths.
We can see that there are 4 groups of 21 in 86 with 2 ones left over. To find the quotient, we need to distribute the 2 ones into the 4 groups. To do this, we first need to decompose the 2 ones into 20 tenths and then put 5 tenths in each group.
Once the 20 tenths are distributed, each group will have 2 tens, 1 one, and 5 tenths, so .
For some division problems, such as or , it is not convenient to draw and reason with base-ten diagrams. We will look at other strategies in upcoming lessons.
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